The \emph{constitutive parameters} are quantities that are
associated with a given media and are used to characterize the
electric and magnetic properties of a material. These are
summarized in table \ref{tbl:cp}.

\begin{table}[!hbp]
\caption{Constitutive Parameters}\label{tbl:cp}
\begin{center}
\begin{tabular}{|c|l|c|}
  \hline
  Symbol & Variable Description & Units\\
  \hline
  $\mu$ & permeability & $H/m$\\
  $\varepsilon$ & permittivity & $F/m$\\
  $\sigma_m$ & magnetic resistivity & $\Omega/m$\\
  $\sigma_e$ & electric conductivity & $S/m$\\
  \hline
\end{tabular}
\end{center}
\end{table}
The \emph{constitutive relationships} are the equations that
relate the \emph{constitutive parameters} to the vector fields
in Maxwell's equations. The four \emph{constitutive relationships}
are the following.
\begin{subequations}
\begin{align}
    \vec{\mathcal{B}}&=\mu\ast\vec{\mathcal{H}}\label{eqn:BH}\\
    \vec{\mathcal{D}}&=\varepsilon\ast\vec{\mathcal{E}}\label{eqn:DE}\\
    \vec{\mathcal{M}_c}&=\sigma_m\ast\vec{\mathcal{H}}\label{eqn:MH}\\
    \vec{\mathcal{J}_c}&=\sigma_e\ast\,\vec{\mathcal{E}}\label{eqn:JE}
\end{align}
\end{subequations}
The \emph{constitutive parameters} are functions of time and the
($*$) operator is a convolution operator. The \emph{constitutive
parameters} are constants in a vacuum and have the following values,
\begin{subequations}
\begin{align}
    &\mu=\mu_0=4\pi\cdot10^{-7}&{(H/m)}\\
    &\varepsilon=\varepsilon_0=\frac{1}{c^2\mu_0}\approx8.854\cdot10^{-12}&(F/m)\\
    &\sigma_m=0&(\Omega/m)\\
    &\sigma_e=0&(S/m)
\end{align}
\end{subequations}
where $c=299792458\;(m/s)$, which is the speed of light in a
vacuum.

In general, the \emph{constitutive parameters} can be a function
of the field strength applied, the direction of the field applied,
the location within the material and frequency. Therefore,
materials characterized by their \emph{constitutive parameters}
can be given any of the following labels.
\begin{description}
    \item[Linear/Non-Linear] Does not/Does depend on the applied field strength.
    \item[Isotropic/Non-Isotropic] Does not/Does depend on the direction of the
    applied field.
    \item[Homogeneous/Non-Homogeneous] Does not/Does depend on the location within the
    material.
    \item[Dispersive/Non-Dispersive] Does not/Does depend on frequency.
\end{description}

Materials are also classified as \emph{dielectrics} (insulators),
\emph{magnetics}, \emph{electrical conductors} and \emph{magnetic
conductors}. The classifications are made when one of the current
densities is predominant. The following definitions are therefore
made.
\begin{description}
    \item[dielectric] When electric displacement current density $(\vec{\mathcal{J}_D}=\frac{\partial\mathcal{\vec{D}}}{\partial t})$, is predominant.
    \item[magnetic] When magnetic displacement current density $(\vec{\mathcal{M}_D}=\frac{\partial\mathcal{\vec{B}}}{\partial t})$, is predominant.
    \item[electric conductor] When electric sink (lossy) current density $(\vec{\mathcal{J}_c}=\sigma_e\ast\vec{\mathcal{E}})$, is predominant.
    \item[magnetic conductor (fictitious)] When magnetic sink (lossy) current density $(\vec{\mathcal{M}_c}=\sigma_m\ast\vec{\mathcal{H}})$, is predominant.
\end{description}
